Geometric Neural Ordinary Differential Equations: From Manifolds to Lie Groups

Wotte, Yannik P.; Califano, Federico; Stramigioli, Stefano

doi:10.3390/e27080878

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Geometric Neural Ordinary Differential Equations: From Manifolds to Lie Groups

by

Yannik P. Wotte

^*

,

Federico Califano

and

Stefano Stramigioli

Robotics and Mechatronics, EEMCS, University of Twente (UT), Drienerlolaan 5, 7522 NB Enschede, The Netherlands

^*

Author to whom correspondence should be addressed.

Entropy 2025, 27(8), 878; https://doi.org/10.3390/e27080878

Submission received: 30 May 2025 / Revised: 13 August 2025 / Accepted: 18 August 2025 / Published: 19 August 2025

(This article belongs to the Special Issue Lie Group Machine Learning)

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Abstract

Neural ordinary differential equations (neural ODEs) are a well-established tool for optimizing the parameters of dynamical systems, with applications in image classification, optimal control, and physics learning. Although dynamical systems of interest often evolve on Lie groups and more general differentiable manifolds, theoretical results for neural ODEs are frequently phrased on

R^{n}

. We collect recent results for neural ODEs on manifolds and present a unifying derivation of various results that serves as a tutorial to extend existing methods to differentiable manifolds. We also extend the results to the recent class of neural ODEs on Lie groups, highlighting a non-trivial extension of manifold neural ODEs that exploits the Lie group structure.

Keywords: neural ordinary differential equations; differential geometry; Lie groups; machine learning; optimal control

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MDPI and ACS Style

Wotte, Y.P.; Califano, F.; Stramigioli, S. Geometric Neural Ordinary Differential Equations: From Manifolds to Lie Groups. Entropy 2025, 27, 878. https://doi.org/10.3390/e27080878

AMA Style

Wotte YP, Califano F, Stramigioli S. Geometric Neural Ordinary Differential Equations: From Manifolds to Lie Groups. Entropy. 2025; 27(8):878. https://doi.org/10.3390/e27080878

Chicago/Turabian Style

Wotte, Yannik P., Federico Califano, and Stefano Stramigioli. 2025. "Geometric Neural Ordinary Differential Equations: From Manifolds to Lie Groups" Entropy 27, no. 8: 878. https://doi.org/10.3390/e27080878

APA Style

Wotte, Y. P., Califano, F., & Stramigioli, S. (2025). Geometric Neural Ordinary Differential Equations: From Manifolds to Lie Groups. Entropy, 27(8), 878. https://doi.org/10.3390/e27080878

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Geometric Neural Ordinary Differential Equations: From Manifolds to Lie Groups

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